Open Access Research article

Joint modelling rationale for chained equations

Rachael A Hughes1*, Ian R White2, Shaun R Seaman2, James R Carpenter34, Kate Tilling1 and Jonathan AC Sterne1

Author Affiliations

1 School of Social and Community Medicine, University of Bristol, Bristol, UK

2 MRC Biostatistics Unit, Institute of Public Health, Cambridge, UK

3 London School of Hygiene and Tropical Medicine, London, UK

4 MRC Clinical Trials Unit, London, UK

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BMC Medical Research Methodology 2014, 14:28  doi:10.1186/1471-2288-14-28

Published: 21 February 2014



Chained equations imputation is widely used in medical research. It uses a set of conditional models, so is more flexible than joint modelling imputation for the imputation of different types of variables (e.g. binary, ordinal or unordered categorical). However, chained equations imputation does not correspond to drawing from a joint distribution when the conditional models are incompatible. Concurrently with our work, other authors have shown the equivalence of the two imputation methods in finite samples.


Taking a different approach, we prove, in finite samples, sufficient conditions for chained equations and joint modelling to yield imputations from the same predictive distribution. Further, we apply this proof in four specific cases and conduct a simulation study which explores the consequences when the conditional models are compatible but the conditions otherwise are not satisfied.


We provide an additional “non-informative margins” condition which, together with compatibility, is sufficient. We show that the non-informative margins condition is not satisfied, despite compatible conditional models, in a situation as simple as two continuous variables and one binary variable. Our simulation study demonstrates that as a consequence of this violation order effects can occur; that is, systematic differences depending upon the ordering of the variables in the chained equations algorithm. However, the order effects appear to be small, especially when associations between variables are weak.


Since chained equations is typically used in medical research for datasets with different types of variables, researchers must be aware that order effects are likely to be ubiquitous, but our results suggest they may be small enough to be negligible.

Chained equations imputation; Gibbs sampling; Joint modelling imputation; Multiple imputation; Multivariate missing data